arXiv:1407.4234v1 [cs.AI] 16 Jul 2014
A Plausibility Semantics for Abstract Argumentation Frameworks
Emil Weydert
Individual and Collective Reasoning Group ILIAS-CSC, University of Luxembourg
Abstract
We propose and investigate a simple plausibility-based extension semantics for abstract argumentation frame-works based on generic instantiations by default knowl-edge bases and the ranking construction paradigm for default reasoning.1
1
Prologue
The past decade has seen a flourishing of abstract argumen-tation theory, a coarse-grained high-level form of defeasible reasoning introduced by Dung [Dung 95]. It is characterized by a top-down perspective which ignores the logical fine structure of arguments and focuses instead on logical (con-flict, support, ...) or extra-logical (preferences, ...) relations between given black box arguments so as to identify reason-able argumentative positions. One way to address the com-plexity of enriched argument structures carrying interacting relations, and to identify the best approaches for evaluating Dung’s basic attack frameworks as well as more sophisti-cated argumentation systems, is to look for deeper unifying semantic foundations allowing us to improve, compare, and judge existing proposals, or to develop new ones.
A major issue is to what extent an abstract account can adequately model concrete argumentative reasoning in the context of a sufficiently expressive, preferably defeasible logic. The instantiation of abstract frameworks by more fine-grained logic-based argument constructions and configura-tions is therefore an important tool for justifying or criti-cising abstract argumentation theories. Most of this work is however based on the first generation of nonmonotonic for-malisms, like Reiter’s default logic or logic programming. While these are closer to classical logic and the original spirit of Dung’s approach, it is well known that they fail to model plausible implication. In fact, they are haunted by counterintuitive behaviour and violate major desiderata for default reasoning encoded in benchmark examples and ra-tionality postulates [Mak 94]. For instance, the only way to deal even with simple instances of specificity reasoning are opaque ad hoc prioritization mechanisms.
1
This is an improved - polished and partly revised - version of my ECSQARU 2013 paper. It adds a link to structured argumen-tation, refines the semantic instantiation concept, and discusses at-tacks between inference pairs.
The goal of the present work is therefore to supple-ment existing instantiation efforts with a simple ranking-based semantic model which interprets arguments and at-tacks by conditional knowledge bases. The well-behaved ranking construction semantics for default reasoning [Wey 96, 98, 03] can then be exploited to specify a new exten-sion semantics for Dung frameworks which allows us to di-rectly evaluate the plausibility of argument collections. Its occasionally unorthodox behaviour may shed a new light on basic argumentation-theoretic assumptions and concepts.
We start with an introduction to default reasoning based on the ranking construction paradigm. After a short look at abstract argumentation theory, we show how to interpret abstract argumentation frameworks by instantiating the ar-guments and characterizing the attacks with suitable sets of conditionals describing constraints over ranking mod-els. Based on the concept of generic instantiations, i.e. us-ing minimal assumptions, and plausibility maximization, we then specify a natural ranking-based extension semantics. We conclude with a simple algorithm, some instructive ex-amples, and the discussion of several important properties.
2
Ranking-based default reasoning
We assume a basic languageL closed under the usual
propo-sitional connectives, together with a classical satisfaction relation |= inducing a monotonic entailment relation ⊢ ⊆ 2L× L. The model sets of (L, |=) are denoted by [[ϕ]] =
{m | m |= ϕ}, resp. [[Σ]] = ∩ϕ∈Σ[[ϕ]] for Σ ⊆ L. BLis the
boolean proposition algebra overBL= {[[ϕ]] | ϕ ∈ L}. Let
Cn(Σ) = {ψ | Σ ⊢ ψ}.
Default inference is an important instance of nonmono-tonic reasoning concerned with drawing reasonable but po-tentially defeasible conclusions from knowledge bases of the formΣ ∪ ∆, where Σ ⊆ L is a set of assumptions or facts,
e.g. encoding knowledge about a specific state of affairs in the domain languageL, and ∆ ⊆ L(։, ❀) is a collection of
conditionals expressing strict or exception-tolerant implica-tional information overL, which is used to guide defeasible
inference.L(։, ❀) = {ϕ ։ ψ | ϕ, ψ ∈ L} ∪ {ϕ ❀ ψ | ϕ, ψ ∈ L} is the corresponding flat conditional language on
top ofL. In the following we will focus on finite Σ and ∆. ∆→ = {ϕ → ψ | ϕ ։ ψ, ϕ ❀ ψ} collects the material
implications corresponding to the conditionals in∆.
impliesψ, forcing us to accept ψ given ϕ. The default
im-plicationϕ ❀ ψ tells us that ϕ plausibly/normally implies ψ, and only recommends the acceptance of ψ given ϕ. The
actual impact of a default depends of course on the context
Σ ∪ ∆ and the chosen nonmonotonic inference concept |∼,
which will be discussed later.
We can distinguish two perspectives in default rea-soning: the autoepistemic/context-based one, and the plausibilistic/quasi-probabilistic one. The former is exem-plified by Reiter’s default logic, where defaults are usually modeled by normal default rules of the formϕ : ψ/ψ (if ϕ,
and it is consistent thatψ, then ψ). A characteristic feature
is that the conclusions are obtained by intersecting suitable equilibrium sets, known as extensions.
The alternative is to use default conditionals interpreted by some preferential or valuational semantics, e.g. System Z [Pea 90, Leh 92], or probabilistic ME-based accounts [GMP 93] (ME = maximum-entropy). For historical reasons and technical convenience (closeness to classical logic), the first approach has received most attention, especially in the con-text of argumentation. However, this ignores the fact that the conditional semantic paradigm has a much better record when it comes to the natural handling of benchmark exam-ples and the satisfaction of rationality postulates [Mak 94]. It therefore seems promising to investigate whether such semantic-based accounts can also help to instantiate and evaluate abstract argumentation frameworks.
Our default conditional semantics for interpreting argu-mentation frameworks is based on the simplest plausibility measure concept able to reasonably handle independence and conditionalization, namely Spohn’s ranking functions [Spo 88, 12], or more generally, ranking measures [Wey 94]. These are quasi-probabilistic belief valuations express-ing the degree of surprise or implausibility of propositions. Integer-valued ranking functions were originally introduced by Spohn to model the iterated revision of graded plain be-lief. We will consider[0, ∞]real-valued ranking measures2,
where∞ expresses doxastic impossibility.
Definition 2.1 (Real-valued ranking measures)
A map R : BL → ([0, ∞], 0, ∞, +, ≤) is called a
real-valued ranking measure iff R([[T]]) = 0, R([[F]]) =
R(∅) = ∞, and for all A, B ∈ BL, R(A ∪ B) =
min≤{R(A), R(B)}. R(.|.) is the associated conditional
ranking measure defined byR(B|A) = R(A ∩ B) − R(A) if
R(A) 6= ∞, else R(B|A) = ∞. R0is the uniform ranking
measure, i.e.R0(A) = 0 for A 6= ∅. If B = BL, we will use
the abbreviationR(ϕ) := R([[ϕ]]).
For instance, the order of magnitude reading interprets rank-ing measure valuesR(A) as exponents of infinitesimal
prob-abilitiesP (A) = pAεR(A), which explains the parallels with
probability theory. The monotonic semantics of our condi-tionals ։, ❀ is based on the satisfaction relation |=rk. The
corresponding truth conditions are
• R |=rkϕ ։ ψ iff R(ϕ ∧ ¬ψ) = ∞.
• R |=rkϕ ❀ ψ iff R(ϕ ∧ ψ) + 1 ≤ R(ϕ ∧ ¬ψ).
2
Although for us, rational values would actually be sufficient.
That is, we assume that a strict implicationϕ ։ ψ states
thatϕ ∧ ¬ψ is doxastically impossible.
Note that we may replaceϕ ։ ψ by ϕ ∧ ¬ψ ❀ F, i.e. it
would be actually enough to considerL(❀). We use ≤ with
a threshold because this provides more discriminatory power and also guarantees the existence of minima for relevant ranking construction procedures. The exchangeability of ar-bitraryr, r′ 6= 0, ∞ by automorphisms allows us to focus,
by convention, on the threshold1. For ∆ ∪ {δ} ⊆ L(։, ❀),
we set
[[∆]]rk= {R | R |=rk ∆}, ∆ ⊢rkδ iff [[∆]]rk ⊆ [[δ]]rk.
⊢rkis monotonic and verifies the axioms and rules of
prefer-ential conditional logic and disjunctive rationality (threshold semantics: no rational monotony) for ❀ [KLM 90].
But it is important to understand that the central con-cept in default reasoning is not some monotonic condi-tional logic for L(։, ❀), but a nonmonotonic meta-level
inference relation |∼ over L ∪ L(։, ❀) specifying which conclusions ψ ∈ L can be plausibly inferred from finite Σ ∪ ∆ ⊆ L ∪ L(։, ❀). We write Σ ∪ ∆ |∼ ψ, or
alter-nativelyΣ |∼∆ψ, and set C∆|∼(Σ) = {ψ | Σ |∼∆ψ}.
The ranking semantics for plausibilistic default reasoning is based on nonmonotonic ranking choice operatorsI which
map each finite ∆ ⊆ L(։, ❀) to a collection I(∆) ⊆ [[∆]]rk of preferred ranking models of ∆. A corresponding
ranking-based default inference notion|∼Ican then be spec-ified by
Σ |∼I∆ψ iff for all R ∈ I(∆), R |=rk∧Σ ❀ ψ.
Similarly, we can also define a monotonic inference concept characterizing the strict consequences.
Σ ⊢I∆ψ iff for all R ∈ I(∆), R |=rk∧Σ ։ ψ.
IfI(∆) = [[∆]]rk,|∼I∆is, modulo cosmetic details,
equiv-alent to preferential entailment (System P) [KLM 90]. If
≤pt describes the pointwise comparison of ranking
mea-sures, i.e. R ≤pt R′ iff for allA ∈ BL R(A) ≤ R′(A),
then I(∆) = {M in≤pt[[∆]]rk} essentially characterizes System Z [Pea 90]. Because these approaches fail to ade-quately deal with inheritance to exceptional subclasses, we introduced and developed the construction paradigm for de-fault reasoning [Wey 96, 98, 03], which is a powerful strat-egy for specifying reasonable I based on Spohn’s
Jeffrey-conditionalization for ranking measures. The resulting de-fault inference notions are well-behaved and show nice in-heritance features. The essential idea is that defaults do not only specify ranking constraints, but also admissible con-struction steps to generate them. In particular, for each de-fault ϕ ❀ ψ, we are allowed to uniformly shift upwards
(make less plausible/increase the ranks of) the ϕ ∧
¬ψ-worlds, which amounts to strengthen belief in the corre-sponding material implication ϕ → ψ. If W is finite, this
is analogous to specifying the rank of a world by adding a weight≥ 0 for each default it violates. More formally, we
define a shifting transformationR → R + r[ρ] such that for
each ranking measureR, χ, ρ ∈ L, and r ∈ [0, ∞], we set (R + r[ρ])(χ) = min{R(χ ∧ ρ) + r, R(χ ∧ ¬ρ)}.
Definition 2.2 (Constructibility)
Let ∆ = {ϕi ❀ / ։ ψi | i ≤ n} ⊆ L(։, ❀). A
ranking measureR′ is said to be constructible fromR over ∆, written R′ ∈ Constr(∆, R), iff there are r
i ∈ [0, ∞]
s.t.R′= R + Σ
i≤nri[ϕi∧ ¬ψi].3
For instance, we obtain a well-behaved robust default infer-ence relation, System J [Wey 96], just by settingIJ(∆) =
Constr(∆, R0) ∩ [[∆]]rk. To implement shifting
minimiza-tion, we may strengthen System J by allowing proper shift-ing (ri> 0) only if the targeted ranking constraint
interpret-ing a defaultϕi ❀ ψi is realized as an equality constraint
R(ϕi ∧ ψi) + 1 = R(ϕi∧ ¬ψi). Otherwise, the shifting
wouldn’t seem to be justified in the first place.
Definition 2.3 (Justifiable constructibility)
R is called a justifiably constructible model of ∆, written R ∈ Ijj(∆) iff R |=rk ∆, R = R0+ Σi≤nri[ϕi∧ ¬ψi],
and for eachrj> 0, R(ϕj∧ ψj) + 1 = R(ϕj∧ ¬ψj).
It follows from a standard property of entropy maximiza-tion (ME) that the order-of-magnitude translamaximiza-tion of ME, in the context of a nonstandard model of the reals with in-finitesimals [GMP 93, Wey 95], to the ranking level always produces a canonical justifiably constructible ranking model
Rme. We set Ime(∆) = {Rme∆ }. Hence, if ∆ 6 ⊢rk F,
R∆
me ∈ Ijj(∆) 6= ∅. If Ijj(∆) is a singleton, we have
therefore|∼jj= |∼me. This holds for instance for minimal
core default sets∆ [GMP 93], where no doxastically
pos-sibleϕi∧ ¬ψi, i.e.∆ 6 ⊢rk ϕi∧ ¬ψi ❀ F, is covered by
otherϕj∧ ¬ψj. However, because of its fine-grained
quan-titative character,|∼meis actually representation-dependent,
i.e. the solution depends on how we describe a problem in L, it is not invariant under boolean automorphisms of BL. Fortunately, there are two other natural
representation-independent ways to pick up a canonical justifiably con-structible model.
• System JZ is based on on a natural canonical
hierarchi-cal ranking construction in the tradition of System Z and ensures justifiable constructibility [Wey 98, 03]. It consti-tutes a uniform way to implement the minimal informa-tion philosophy at the ranking level.
• System JJR is based on the fusion of the justifiably
con-structible ranking models of∆, i.e. Ijjr(∆) = {R∆jjr},
where for allA ∈ BL,R∆jjr(A) = M in≤ptIjj(∆). |∼
jjr
may be of particular interest because its canonical rank-ing model is at least as plausible as every justifiably con-structible one.
Note that for non-canonical Ijj(∆), it is possible that
R∆
jjr 6∈ Ijj(∆). We have |∼jj ⊂ |∼me, |∼jz, |∼jjr.
Fortu-nately, for the generic default sets we will use to interpret abstract argumentation frameworks, all four turn out to be equivalent. To conclude this section, let us consider a simple example with a single JJ-model.
Big birds example:
Non-flying birds are not inferred to be small.
3
Similar ideas can be found in [BSS 00, KI 01].
{B, ¬F } ∪ {B ❀ S, B ❀ F, ¬S ❀ ¬F } 6|∼jj S
The canonical JJ/ME/JZ/JJR-model is then
R = R0+ 1[¬F ] + 2[¬S ∧ F ]. But R 6|=rk B ∧ ¬F ❀ S
becauseR(B ∧ ¬F ∧ S) = R(B ∧ ¬F ∧ ¬S) = 1
3
Abstract argumentation
The idea of abstract argumentation theory, launched by Dung [Dun 95], has been to replace the traditional bottom-up strategy, which models and exploits the logical fine struc-ture of arguments, by a top-down perspective, where argu-ments become black boxes evaluated only based on knowl-edge about specific logical or extra-logical relationships connecting them. It is interesting to see that such a coarse-grained relational analysis often seems sufficient to deter-mine which collections of instantiated arguments are reason-able. In addition to possible conceptual and computational gains, the abstract viewpoint offers furthermore a powerful methodological tool for general argumentation-theoretic in-vestigations.
An abstract argumentation framework in the original sense of Dung is a structure of the form A = (A, ✄),
whereA is a collection of abstract entities representing ar-guments, and ✄ is a possibly asymmetric binary attack re-lation modeling conflicts between arguments. To grasp the expressive complexity of real-world argumentation, several authors have extended this basic account to include further inferential or cognitive relations, like support links, prefer-ences, valuations, or collective attacks. Our general defini-tion4[Wey 11] for the first-order context is as follows.
Definition 3.1 (Hyperframeworks) A general abstract
ar-gumentation framework, or hyperframework (HF), is just a structure of the formA = (A, (Ri)i∈I, (Pj)j∈J), where A
is the domain of arguments, theRi are conflictual, and the
Pi non-conflictual relations over A. B ⊆ A is said to be
conflict-free iff it does not instantiate a conflictual relation. For instance, standard Dung frameworks(A, ✄) carry one
conflictual and no non-conflictual relations. The general in-ferential task in abstract argumentation is to identify reason-able evaluations of the arguments described byA, e.g. to find
out which sets of arguments describe acceptable argumen-tative positions. These are called extensions. In Dung’s sce-nario, the extensions areE ⊆ A obeying suitable
acceptabil-ity conditions in the context ofA, the minimal requirement
being the absence of internal conflicts. For instance,E is
ad-missible iff it is conflict-free and each attacker of ana ∈ E
is attacked by someb ∈ E. E is grounded/preferred iff it
is minimally/maximally admissible, it is stage iffE ∪ ✄′′E
is maximal, semi-stable if it is also admissible, and stable iff A − E = ✄′′E. Here ✄′′E is the relational image of
E, i.e. the set of a ∈ A attacked by some b ∈ E. In
con-crete decision contexts, we may however also want to ex-ploit finer-grained assessments of arguments, like prioritiza-tions or classificaprioritiza-tions. This suggests a more general seman-tic perspective [Wey 11].
Definition 3.2 (Hyperextensions) A hyperframework
se-mantics is a mapE associating with each hyperframework
4
A = (A, (Ri)i∈I, (Pj)j∈J) of a given signature a
collec-tionE(A) of distinguished evaluation structures expanding A, of the form (A, InA, (F
h)h∈H). InAis here a
conflict-free subset ofA. The elements of E(A) are called hyperex-tensions ofA.
InAplays here the role of a classical extension, whereas the
Fh(h ∈ H) express more sophisticated structures over
ar-guments, e.g. a posteriori plausibility orderings, value predi-cates, or completions of framework relations considered par-tial. IfH = ∅, we are back to Dung.
4
Concretizing arguments
Ideally, abstract argumentation frameworks should be reconstructible as actual abstractions of logic-based argu-mentation scenarios. Such an anchoring seems required to develop, evaluate, and apply the abstract models in a suitable way. In a first step, this amounts to instantiate the abstract arguments from the framework domain by logical entities representing concrete arguments, and to interpret the abstract framework structure by specific inferential or evaluational relationships fitting the conceptual intentions the abstract level tries to capture. In what follows we will sketch a natural hierarchy of instantiation layers, passing from more concrete, deep instantiations, to more abstract, shallow ones, with a focus on the intermediate level.
Structured instantiations:
We start with logic-based structured argumentation over a defeasible conditional logicLδ = (L ∪ L(։, ❀), ⊢δ, |∼δ),
with(L, ⊢) as a classical Tarskian background logic. For the
moment, we do not impose any further a priori conditions onLδ. But eventually we will turn to specific ranking-based
default formalisms. In the context ofLδ, a concrete
defeasi-ble argumenta for a claim ψa ∈ L, exploiting some given
general knowledge baseΣ ∪ ∆, is modeled by a finite rooted
defeasible inference tree Ta whose nodes s are tagged by
local claimsηs∈ L ∪ L(։, ❀) such that
• the root node is tagged by ψa,
• the leaf nodes are tagged by ηs ∈ Σa∪ ∆a∪ Λ, where
Λ = {T}∪{ϕ ։ ϕ, ϕ ❀ ϕ | ϕ ∈ L} (basic tautologies), • the non-leaf nodes are tagged by ηs ∈ L s.t. Γs |∼δ ηs
whereΓsis the set of claims from the children ofs.
Σa∪ ∆ais the contingent premise set ofa, the premises
be-ing the claims of the leaf nodes. Within concrete arguments, the local justification steps, e.g. fromΓstoηs, are typically
assumed to be elementary, like instances of modus ponens. To handle reasoning by cases, which holds for plausible im-plication, we may also apply the disjunctive modus ponens for ։ and ❀, e.g.
Γs= {ϕ1∨ . . . ∨ ϕn, ϕ1❀ψ1, . . . , ϕn ։ ψn} |∼δ ψ
1∨ . . . ∨ ψn(= ηs).
If Γs ⊆ L ∪ L(։), we can replace |∼δ by ⊢δ and
ob-tain a strict inference step. For our purposes we may ig-nore the exact nature of the justification steps. Note that the correctness of local inference steps does not entail the
global correctness of the argumenta. Consider for instance Σa∪ ∆a = {ϕ} ∪ {ϕ ❀ ψ, ψ ❀ ¬ϕ}, which is consistent
w.r.t.|∼δ= |∼I.
{ϕ} ∪ {ϕ ❀ ψ} |∼δ ψ and {ψ} ∪ {ψ ❀ ¬ϕ} |∼δ ¬ϕ,
butΣa ∪ ∆a 6|∼δ ¬ϕ. This example looks odd because
ac-cepting the whole argument would require the acceptance of all its claims, which is blocked byϕ, ¬ϕ ⊢ F. In fact, a
natural requirement for an acceptable argumenta would be
Material consistency: Σa∪ ∆→a 6 ⊢ F.
This means that the factual premises and the material impli-cations corresponding to the conditional premises are classi-cally consistent. Note that this condition is strictly stronger than Σa ∪ ∆a 6|∼δ F because we typically have {T ❀
ϕ, ¬ϕ} 6|∼δ F whereas {T → ϕ, ¬ϕ} ⊢ F. However, in
practice, without omniscience w.r.t. propositional logic, it may not be clear whether these global conditions are actu-ally satisfied. Real arguments may well be inconsistent in the strong sense.
In structured argumentation, an argument tree has two functions: first, to describe and offer a prima facie justifica-tion for a claim, and secondly, to specify target points where other arguments may attack. It is essentially a computational tool which is intended to help identifying - or even defining - inferential relationships within a suitable defeasible con-ditional logicLδ, and to help specifying attack relations to
determine reasonable argumentative positions.
But what can we say about the semantic content of an argument represented by such a tree? What is an agent committed to if he accepts or believes a given argument, or a whole collection of arguments? Which tree attributes have to be known to specify this content? What is the meaning of attacks between arguments?
Conditional instantiations:
Our basic idea is that, whatever the requirements for argu-mentation trees in the context ofLδ, and whatever the
con-tent of an argumenta represented by such a tree Ta, it should
only depend on the collection of local claims{ηs | s node
of Ta}, and more specifically, on the choice of the main
claimψa, the premise claimsΣa∪ ∆a, and the intermediate
claims Ψa. In fact, because the acceptance of a structured
argument includes the acceptance of all its subarguments, we have to consider the main claims of the subarguments as well. So we can assume that the content of Ta is fixed
by the triple(Σa∪ ∆a, Ψa, ψa). An agent accepting a
ob-viously has to be committed to all the elements of the base
Σa∪ ∆a∪ Ψa∪ {ψa}.
To be fully acceptable w.r.t.Lδ, the structured argument
also has to be globally correct in the sense that all its lo-cal claims are actually defeasibly entailed by Σa∪ ∆a. In
particular,Σa∪ ∆a |∼δ ψ for each ψ ∈ Ψa∪ {ψa}. This
requirement should also hold for each subargumentb of a.
But note that, because of defeasibility, this does not exclude that the premisesΣb∪ ∆bof a subargumentb could
implic-itly infer the negation of a local claimψxexternal tob, as
long as this conflicting inference is eventually overridden by the full premise set Σa ∪ ∆a. It follows that the
undermine global correctness. But if the intermediate claims are always inferred and therefore implicitly present, we may actually dropΨaand just consider for each globally correct
argumenta the finite inference pair (Σa∪ ∆a, ψa).
Given a pure Dung frameworkA = (A, ✄) and the de-feasible conditional logicLδ, a structured instantiationIstr
ofA maps each a ∈ A to a globally correct argument tree
Ta overLδ. On the most general level, we do not want to
impose a priori further restrictions beyond inferential cor-rectness. In practice one may however well decide to focus on specific argument trees, e.g. those using specific justifi-cation steps. EachIstr(a) specifies a correct inference pair
Ilog(a) = (Σa∪ ∆a, ψa), which we call a conditional
log-ical instantiation of a over Lδ.Ilog specifies the intended
logical content of an argument on the syntactic level. Note that it depends on the tree concept whether we can obtain all the correct inference pairs.
In monotonic argumentation, the consistency and mini-mality of the premise sets are standard assumptions. But within defeasible argumentation, a more liberal perspective may be preferable. For instance, on the structured level, we want to allow arguments claiming F. The reductio ad absur-dum principle then offers a possibility to attack arguments from within. Consequently, we also have to accept instan-tiating inference pairs whose conclusion is F. On the other hand, material consistency, the existence of models of Σa
which do not violate any conditional in∆a, is a natural
re-quirement in the context of argumentation theory. But we can replace it by a qualified version, restricted to those in-stances whereΣa∪ ∆ais actually consistent.
What about minimality? First, it may obviously fail for inference pairs obtained by flattening argument trees. Of course, we could consider an additional minimization step where we replace each(Σa∪ ∆a, ψa) by all those (Φ, ψa)
withΦ ⊆ Σa∪ ∆a and which are minimal s.t.Φ |∼δ ψa.
Although this may be computationally costly, it could be theoretically appealing. However, minimality could also be questioned because by adding premises, a conclusion may successively get accepted, rejected, and accepted again, letting the character of the inferential support change between different levels of specifity, which calls for a discrimination between the corresponding inference pairs. Proponents of minimality object that these types of support could, perhaps, also be reproduced by suitable minimal
(Φ, ψa). However, this assumption is not sustainable for
ranking-based semantics for argumentation, because here the results may change if we restrict ourselves to minimal premise sets. In fact, shrinkingΣa∪ ∆atoΦ may actually
increase the set of possible attacks. In particular, we could have attacks on all the minimal Φ |∼δ ψ
a, but none on
Σa∪ ∆a|∼δψa. Hence premise minimality may fail.
Shallow instantiations:
Let us recall our task: exploiting a ranking semantics for default reasoning to provide a plausibilistic semantics for abstract argumentation. But inference pairs, which pop-ulate the conditional logical instantiation level, are still rather complex and opaque objects. To model argumentation frameworks and their semantics, we would here have to deal
with sets of sets of conditionals, whose inferential interac-tions may furthermore be hard to assess. We therefore prefer to start with simpler entities and to seek more abstraction.
Consider the main goal of an agent: to extract from ar-gument configurations suitable beliefs, expressed in the do-main language L, whose plausibility is semantically
mod-eled by ranking measures over BL. Given an inference
pair (Σa ∪ ∆a, ψa) representing the full conditional
log-ical content of an argument a, in addition to the main
claim ψa, there are three relevant collections of formulas:
Σa, C⊢(∆a∪, Σa), C|∼(∆a∪ Σa) which represent resp. the
premises, the strict, and the defeasible consequences. If the language is finitary, this gives us fourL-formulas
represent-ing the relevant propositionalL-content. • ϕa= ∧C⊢(Σa) (premise content).
• θa = ∧C⊢(∆a∪ Σa) (strict content).
• δa= ∧C|∼(∆a∪ Σa) (defeasible content).
• ψa(main claim).
We haveδa ⊢ θa ⊢ ϕa, andδa ⊢ ψa by inferential
correct-ness.δa specifies the strongest possible claim based on the
information made available by the argument. For our seman-tic modeling purposes, we will assume thatψa = δa. If we
abstract away from the representational details, we arrive at our central concept: the shallow semantic instantiation ofa
extracted from the conditional logical instantiationIlog(a).
Isem(a) = ([[ϕa]], [[θa]], [[ψa]]).
In the following, we will sloppily denote Isem(a) by
(ϕa, θa, ψa). One should emphasize that these propositional
semantic profiles are not intended to grasp the full nature of arguments, but only to reflect certain characteristics ex-ploitable by suitable argumentation semantics. We observe that each proposition triple (ϕ, θ, ψ) with ψ ⊢ θ ⊢ ϕ
can become a shallow instantiation. In fact, if Ilog(a) =
({ϕ, ϕ ։ θ, ϕ ❀ ψ}, ψ), for standard |∼δ, we obtain
Isem(a) = (ϕ, θ, ψ). In terms of ranking constraints, this
gives usR(ϕ ∧ ¬θ) = ∞ and R(ϕ ∧ ψ) + 1 ≤ R(ϕ ∧ ¬ψ).
5
Concretizing attacks
One argument attacks another argument if accepting the first interferes with the inferential structure or goal of the second one. To avoid a counterattack, the premises of the attacked argument should also not affect the inferential success of the attacker, otherwise the presupposition of the attack could be undermined. In the following we will investigate attack rela-tions between conditional logical resp. shallow semantic in-stantiations of abstract arguments. We start with the former. LetIlog(a) = (Σa∪ ∆a, ψa) and Ilog(b) = (Σb∪ ∆b, ψb)
be two correct inference pairs forLδ. We distinguish two
scenarios: unilateral and mutual attack. The idea is to say that (Σa ∪ ∆a, ψa) unilaterally attacks (Σb ∪ ∆b, ψb) iff
the premises of both arguments together withψaenforce the
strict rejection ofψb, i.e.
Σb∪ ∆b∪ Σa∪ ∆a∪ {ψa} ⊢δ ¬ψb,
whereas the defeasible inference ofψafrom the premises is
Σa∪ ∆a∪ Σb∪ ∆b|∼δ ψa.
On the other hand,(Σa∪ ∆a, ψa) and (Σb∪ ∆b, ψb) attack
each other iff they strictly reject each other’s claims, i.e.
Σb∪ ∆b∪ Σa∪ ∆a∪ {ψa} ⊢δ¬ψb, and
Σa∪ ∆a∪ Σb∪ ∆b∪ {ψb} ⊢δ ¬ψa.
This holds for instance if their premise sets, resp. their claims, are classically inconsistent. This definition provides one of the strongest possible natural attack relations for in-ference pairs. Note that we have a self-attack iff the premise set is inconsistent, i.e.Σa∪ ∆a ⊢δ F. To exploit the
pow-erful semantics of ranking-based default reasoning, in what follows we will assume that|∼δ= |∼I, whereI is a ranking
choice function.
How can we exploit the above approach to define attacks between shallow instantiations, e.g.Isem(a) = (ϕa, θa, ψa)
andIsem(b) = (ϕb, θb, ψb)? The corresponding inference
pairs areIlog(a) = ({ϕa, ϕa ։ θa, ϕa ❀ ψa}, ψa) and Ilog(b) = ({ϕb, ϕb ։ θb, ϕb ❀ψb}, ψb). For an unilateral
attack fromIlog(a) on Ilog(b), we must have
I(∆a∪ ∆b) |=rk ϕa∧ ϕb∧ ψa։ ¬ψb, and
I(∆a∪ ∆b) |=rkϕa∧ ϕb❀ψa.
This is, for instance, automatically realized ifψa ⊢ ¬ψb,
ϕa ⊢ ϕb, and we have logical independence elsewhere. For
a bilateral attack, we may just drop the conditionϕa ⊢ ϕb.
However, we do not have to presuppose that all the attacks result from the logical structure induced by the instantiation. In fact, in addition to the instantiation-intrinsic attack rela-tionships, there could be further attack links derived from a separate conditional knowledge base reflecting other known attacks.
From a given Dung frameworkA = (A, ✄) and a shallow
instantiation I = Isem, if we adopt the ranking semantic
perspective and the above attack philosophy, we can induce a collection of conditionals specifying ranking constraints. For anya ∈ A, the shallow inference pair supplies ϕa։ θa
(alternatively,ϕa ∧ ¬θa ❀ F) and ϕa ❀ ψa. For every
attacka ✄ b, we get at least ψa∧ ψb❀F. Note that this is a
consequence of choosing maximal claims at the instantiation level. For each unilateral attacka✄b we must add ϕa∧ϕb❀ ψato preserve the inferential impact ofa in the context of b.
The resulting default base is
∆A,I = {ϕ
a❀ψa, ϕa։ θa | a ∈ A} ∪ {ψa∧ ψb❀F | a ✄ b or b ✄ a}
∪ {ϕa∧ ϕb❀ψa| a ✄ b, b 6 ✄ a}.
We observe that for each 1-loop, we get ψa ❀ F and ϕa ❀ψa, hence∆A,I ⊢I ¬ϕ
a. The doxastic impossibility
ofϕa illustrates the paradoxical character of self-attacking
arguments. The belief states compatible with an instantiated frameworkA, I are here represented by the ranking models
of∆A,I.
Conversely, we can identify for each instantiationI of A
and each collection of ranking measures R |=rk ∆A,I =
{ϕa ❀ψa, ϕa ։ θa | a ∈ A} the attacks supported by all
theR ∈ R. Let ✄R
I be the resulting attack relation, that is,
for eacha, b ∈ A
a ✄RI b iff for all R ∈ R, R |=rk ψa∧ ψb❀F and
(R |=rkϕa∧ ϕb❀ψaorR 6|=rkϕa∧ ϕb ❀ψb). The second disjunct is the result of an easy simplification. Ifa or b are self-reflective, we have a ✄RI b because
con-ditionals always hold if the premises are doxastically im-possible. Because in this paper we will mainly consider canonical ranking choice functions, we are going to focus onR = {R}, setting ✄R
I = ✄RI .
Definition 5.1 (Ranking instantiation models)
Let the notation be as usual andA+ = {a ∈ A | a 6✄ a}.
(R, I) is called a ranking instantiation model (more
slop-pily, a ranking model) ofA iff
R |=rk∆A,I = {ϕa❀ψa, ϕa ։ θa| a ∈ A},
and for all a, b ∈ A+, a ✄ b iff a ✄R
I b. Let RA be the
collection of all the ranking instantiation models ofA.
That is, over the non-loopy arguments, the semantic-based attack relation ✄RI specified byR, I has to correspond
ex-actly to the abstract attack relation ✄. The collection of rank-ing instantiation models is not meant to change if we add or drop attack links between self-reflective and other arguments because the details are absorbed by the impossible joint con-texts. IfA and A′share the same 1-loops and the same attack
structure over the other arguments,RA= RA′ .
It also important to observe, and we will come back to this, that eachA = (A, ✄) admits many ranking instantia-tion models(R, I), obtained by varying the ranking values
or the proposition triples associated with the abstract argu-ments.
What can we say about classical types of attack? If we fo-cus on the actual semantic content, rebuttal is characterized by incompatible defeasible consequents, and undermining by a defeasible consequent conflicting with an antecedent. In the ranking context, these two types of attacks can be mod-eled by constraints expressing necessities. The ranking char-acterizations are as follows. Recall thatψa ⊢ ϕa,ψb⊢ ϕb.
a rebuts b: R(ψa∧ ψb) = ∞, e.g. if ψa ⊢ ¬ψb.
a undermines b: R(ψa∧ ϕb) = ∞, e.g. if ψa⊢ ¬ϕb.
In our simple semantic reading, undermining entails rebut-tal becauseψb ⊢ ϕb. There are four qualitative attack
con-figurations involving two arguments:ϕa ∧ ϕb being
com-patible with neither, one, or both of ψa, ψb. If a
asym-metrically undermines b, we have R(ψa ∧ ϕb) = ∞ and
R(ψb∧ ϕa) 6= ∞, hence R(ϕa ∧ ϕb) 6= ∞. This implies
R |=rk ϕa∧ ϕb∧ ψb ։ ¬ψa andR 6|=rk ϕa∧ ϕb ❀ψa, i.e.b ✄R
I a and a 6✄RI b according to our attack semantics. It
follows that undermining has no obvious ranking semantic justification if we stipulate that the defeasible claim entails the antecedent. Also note that rebuttal is entailed by sym-metric and asymsym-metric attacks.
6
Ranking extensions
Ranking instantiation models offer new possibilities to iden-tify reasonable argumentative positions. Let (R, I) be a
model of the framework A = (A, ✄). In the context of
S ⊆ A are coherent premises, i.e. the doxastic possibility
of the joint strict contentsθS = ∧a∈S(ϕa ∧ θa) w.r.t. R,
which meansR(θS) 6= ∞. This excludes self-attacks, but
not conflicts withinS. S = ∅ is by definition coherent
be-cause θ∅ = T. Given that evidence ϕa should not be
re-jected without good reasons, the maximally coherentS ⊆ A
are of particular interest and constitute suitable background contexts when looking for extensions. EachE ⊆ S then
specifies a proposition
ψS,E:= θS ∧ ∧a∈Eψa ∧ ∧a∈A−E¬ψa.
ψS,Echaracterizes those worlds verifying the strict content
of the a ∈ S and exactly the defeasible content of the a ∈ E. Because a ✄R
I b implies R(ψa∧ ψb) = ∞, any
con-flicta ✄ b in E makes ψS,Eimpossible. Note however that
R(ψS,E) = ∞ may also result from non-binary conflicts
in-volving multiple arguments, or a specific choice of logically dependentϕa, ψa. What are the most reasonable extension
candidatesE ⊆ S ⊆ A according to (R, I)? One idea is
to focus on thoseE which induce the most plausible ψS,E
relative toθSamong all their maximal coherent supersetsS.
Definition 6.1 (Ranking extensions) Let(R, I) be a
rank-ing instantiation model ofA = (A, ✄). E ⊆ A is called
a ranking-extension ofA w.r.t. (R, I) iff there is a maximal
coherentS ⊆ A with E ⊆ S and R(ψS,E|θS) = 0.
Observe that ifS = ∅ is the maximally coherent subset of A, then R(ϕa) = ∞ for each a ∈ A and E = ∅ is the only
ranking extension. While the above definition looks rather decent, a cause of concern may be the great diversity of rank-ing models(R, I) available for any given A. Consider for
instanceA = ({p, q, r}, {(p, q), (q, r)}), i.e. p ✄ q ✄ r. A
together with a shallow instantiationI then induces ranking
constraints described by the conditionals in
∆A,I = {ψ
p∧ ψq ❀F, ψq∧ ψr❀F, ϕp∧ ϕq ❀ψp,
ϕq∧ ϕr❀ψq, ϕp❀ψp, ϕq ❀ψq, ϕr❀ψr}.
If we assume that each set{ϕx, ψx} is logically independent
from all the other{ϕy, ψy}, then ∆A,I admits a unique
jus-tifiably constructible model, which therefore automatically must be the JZ- and JJR-model:RA,Ijz . In this example it is obtained by minimally shifting the violation areas of the conditionals.
RjzA,I = R0+∞[ψp∧ψq]+∞[ψq∧ψr]+1[ϕp∧ϕq∧¬ψp]+
1[ϕq∧ϕr∧¬ψq]+1[ϕp∧¬ψp]+1[ϕq∧¬ψq]+1[ϕr∧¬ψr].
Given that S = A is coherent, there are eight
ex-tension candidates. For the doxastically possible alterna-tives, RA,Ijz (ψA,{p,r}) = 2 < 3 = RA,Ijz (ψA,{p}) =
RjzA,I(ψA,{q}) < 4 = RjzA,I(ψA,{r}) < 5 = RjzA,I(ψA,∅) <
∞. Because RA,Ijz (ϕS) = 2, we get RA,Ijz (ψA,{p,r}|ϕS) = 0.
The unique ranking extension is therefore{p, r}, which is
also the standard Dung solution.
However, without any further constraints on the exten-sion generating ranking instantiation model(R, I), we could
pick up as an alternativeR = RA,Ijz + ∞[ψp∧ ψr ∧ ϕq]
such that R(ψp ∧ ψr ∧ ϕq) = ∞, resp. an I enforcing
ψp∧ ψr ∧ ϕq ⊢ F . In both scenarios, the minima would
then becomeR(ψA,{p}) = R(ψA,{q}) = 3, imposing the
ranking extensions{p}, {q}. Because of R(ψA,{p,r}) = ∞,
the standard extension{p, r} would necessarily be rejected.
But this violates a hallmark of argumentation, namely the unconditional support of unattacked arguments, likep. This
shows that we have to control the choice of ranking instan-tiation models to implement a reasonable ranking extension semantics.
The idea is now to choose on one hand, as our doxas-tic background, a well-justified canonical ranking measure model of the default base ∆A,I, e.g. the JZ-model RA,I
jz ,
and on the other hand, implementing Ockham’s razor, the simplest instantiations of the given frameworkA. In
partic-ular, we stipulate that the instantiations of individual argu-ments should by default be logically independent. We em-phasize that the goal here is just to interpret abstract argu-mentation frameworks with a minimal amount of additional assumptions, not to adequately model specific real-world ar-guments.
We can satisfy these desiderata by using disjoint vocabu-laries for instantiating different abstract arguments, and by relying on elementary instances of the defeasible modus po-nens for the corresponding inference pairs. That is, we in-troduce for each a ∈ A independent propositional atoms Xa, Ya, and set Ilog(a) = ({Xa} ∪ {Xa ❀ Ya}, Ya).
The corresponding shallow semantic instantiation is then
I(a) = Isem(a) = (ϕa, ϕa, ψa) = (Xa, Xa, Xa ∧ Ya).
We callI a generic instantiation. Up to boolean isomorphy,
it is completely characterized by the cardinality ofA. If we fix a generic instantiationI, then the unique
justifi-ably constructible ranking model of∆A,I is (a ✁/✄ b: a ✄ b
orb ✄ a) RA
jz = R0+ Σa6✄a1[ϕa∧ ¬ψa] + Σa✄a∞[ϕa∧ ¬ψa] +
Σa✄b1[ϕa∧ ϕb∧ ¬ψa] + Σa✁/✄b∞[ψa∧ ψb]
= R0+ Σa6✄a1[Xa∧ ¬Ya] + Σa✄a∞[Xa∧ ¬Ya] +
Σa✄b1[Xa∧ Xb∧ ¬Ya] + Σa✁/✄b∞[Xa∧ Ya∧ Xb∧ Yb].
Because the{Xa, Ya} are logically independent for distinct
a, and the defaults expressing an attack a ✄ b just concern Xa ∧ Xb, only those Xa with a ✄ a become impossible.
In fact,{ϕa ❀ ψa, ψa ∧ ψa ❀ F} ⊢rk ϕa ❀ F. Hence,
in line with intuition, the ranking instantiation model
(RjzA, I) will trivialize exactly the self-defeating arguments.
Assuming genericity,A+ = {a ∈ A | a 6✄ a} is then the
unique maximal coherent subset ofA. We are now ready to
specify our ranking-based evaluation semantics. Note that all the genericI are essentially equivalent.
JZ-evaluation semantics (JZ-extensions):
Ejz(A) = {E ⊆ A | E ranking extension w.r.t. (RA,Ijz , I)
for any/all genericI}.
There is actually a simple algorithm to identify the JZ-extensions using extension weights.
Definition 6.2 (Extension weight) For each
argumenta-tion frameworkA = (A, ✄), the extension weight function
rA(E) = |A+− E| + |{a ∈ A+− E | ∃b ∈ A+(a ✄ b ∧
b 6 ✄a)}|, if not, rA(E) = ∞.
It is not too difficult to see that rA(E) = RA,Ijz (ψA+,E). Hence,E ∈ Ejz(A) iff rA(E) = min{rA(X) | X ⊆ A}.
That is, the JZ-extensions are those where the sum of the number of non-reflective non-extension arguments and the number of one-sided attacks starting from them is minimal.
7
Examples and properties
To get a better understanding of the ranking extension semantics and its relation with other extension concepts, let us first take a look at how it handles some basic ex-amples. Because of its uncommon semantic perspective and its partly quantitative character, we will observe some unorthodox behaviour. Under instantiation genericity, it is enough to compareRA,I(ψ
A+,E) for E ⊆ A+, or to focus on 1-loop-free frameworks. For each instance, we specify the domainA and the full attack relation ✄. ψA+,{x
1...xn}is abbreviated byψx1,...,xnresp.ψ∅.
Simple reinstatement: {a, b, c} with a ✄ b ✄ c.
The grounded extension {a, c} is the canonical result
put forward by any standard acceptability semantics. The unique JJ-model, i.e. the JZ-model R of ∆A,I, satisfies
R(ψa) = R(ψb) = 3, R(ψc) = 4, R(ψa,c) = 2, and
R(ψ∅) = 5. The other candidates all get rank ∞. Because
R(ψa,c) is minimal, {a, c} is the only JZ-ranking extension,
i.e.Ejz(A) = {{a, c}}.
3-loop: {a, b, c} with a ✄ b ✄ c ✄ a.
Semantics under the admissibility dogm reject
{a}, {b}, {c}, only ∅ is admissible. But the JZ-model R verifies R(ψa) = R(ψb) = R(ψc) = 4 < 5 = R(ψ∅).
Because all the alternatives are set to ∞, our ranking
extensions are the maximal conflict-free sets{a}, {b}, {c},
i.e.,Ejzclearly violates admissibility.
Attack on 2-loop: {a, b, c} with a ✄ b ✄ c ✄ b.
We have R(ψ∅) = 4, R(ψa) = 2, R(ψb) = R(ψc) =
3, R(ψa,c) = 1, but ∞ for the others. Here Ejz(A) =
{{a, c}} picks up the canonical stable extension.
Attack from 2-loop: {a, b, c} with b ✄ a ✄ b ✄ c.
We getR(ψ∅) = 4, R(ψa) = 3, R(ψb) = 2, R(ψc) = 3,
R(ψa,b) = R(ψb,c) = ∞, and R(ψa,c) = 2.
Ejz(A) = {{b}, {a, c}} thus collects the stable
exten-sions.
3,1-loop: {a, b, c} with a ✄ a ✄ b ✄ c ✄ a.
E = ∅ is here the only admissible extension. The
maximal coherent set is A+ = {b, c}, and we get R(ψb) = 1, R(ψc) = 2, as well as R(ψ∅) = 3. It follows
thatEjz(A) = {{b}}, rejecting the stage extension {c}.
3,2-loop: {a, b, c} with b ✄ a ✄ b ✄ c ✄ a.
We have R(ψ∅) = 5, R(ψa) = 4, R(ψb) = 3, and
R(ψc) = 3, i.e. Ejz(A) = {{b}, {c}}. But the stable
extension{b} is the only admissible ranking extension.
The previous examples show that the ranking extension semantics Ejz can diverge considerably from all the other
major proposals found in the literature. It may look as if the main difference is its more liberal attitude towards some non-admissible, but still justifiable extensions. However, the semantics has an even more exotic flavour. Consider the following scenarios, where we indicate the minimal extension weightsrA(E).
2-loop chain: {a, b, c}, b ✄ a ✄ b ✄ c ✄ b : r({a, c}) = 1 < 2 = r({b}).
Splitted 3-chain: {a, b, c, d}, a ✄ b ✄ c, a ✄ d ✄ c : r({a, c}) = r({b, d}) = 4.
Spoon: {a, b, c, d}, a ✄ b ✄ c ✄ d ✄ c : r({a, d}) = r({a, c}) = r({b, d}) = 3.
The first example documents the rejection of a stable extension, namely {b}. The second one illustrates the
impact of quantitative considerations when dealing with a splitted variant of simple reinstatement. The third instance shows the coexistence of two stable extension with a non-admissible one. That is, even attack-free a can be
questioned under certain circumstances. It follows that the above ranking semantic interpretation of argumentation frameworks deviates considerably from standard accounts and expectations. Let us now investigate how Ejz handles
some common principles for extension semantics.
Isomorphy:f : A ∼= A′ implies f′′: E(A′) ∼= E(A).
Conflict-freedom: Ifa, b ∈ E ∈ E(A), then a 6 ✄ b.
CF-maximality: If E ∈ E(A), then E is a maximal
conflict-free subset ofA.
Inclusion-maximality: IfE, E′∈ E(A) and E ⊆ E′, then
E = E′.
Reinstatement: IfE ∈ E(A), a ∈ A, and for each b ✄ a
there is ana′ ∈ E with a′✄b, then a ∈ E.
Directionality: Let A1 = (A1, ✄1), A2 = (A2, ✄2)
be such that A1 ∩ A2 = ∅, ✄0 ⊆ A1 × A2,
A = (A1 ∪ A2, ✄1 ∪ ✄0 ∪ ✄2). Then we have
E(A1) = {E ∩ A1| E ∈ E(A)}.
Theorem 7.1 (Basic properties)
Ejzverifies isomorphy, conflict-freedom, inclusion
maximal-ity, and CF-maximality. It falsifies reinstatement and direc-tionality.
The first four features are easy consequences of the Ejz
-specification. The violation of reinstatement directly fol-lows from how the semantics handles 3-loops. The spoon example documents the failure of directionality if we set
A1 = {a, b}. But directionality also fails for other
promi-nent approaches, like the semi-stable semantics. Note how-ever that it can be indirectly enforced by usingEjz as the
The following properties are inspired by the cumulativ-ity principle for nonmonotonic reasoning. They state that if we drop an argument rejected by every extension, then this shouldn’t add or erase skeptically supported arguments.
Rejection cumulativity: (A|B: A restricted to B)
Rej-Cut: Ifa 6∈ ∪E(A), then ∩E(A|A − {a}) ⊆ ∩E(A).
Rej-CM: Ifa 6∈ ∪E(A), then ∩E(A) ⊆ ∩E(A|A − {a}).
Although our semantics relies on default inference notions verifying cumulativity at the level of |∼I∆, it nevertheless
fails to validate the previous postulates.
Theorem 7.2 (No rejection cumulativity)
Ejzviolates Rej-Cut and Rej-CM.
The counterexample for Rej-CUT is provided byb ✄ c ✄ a ✄ b ✄ a, because {b} 6⊆ {b} ∩ {c}. The one for Rej-CM is
obtained by addingc ✄ b. Here {c} 6⊆ {b} ∩ {c}.
Another idea for combining plausibilistic default reason-ing and argumentation theory has been presented in [KIS 11]. It combines defeasible logic programming with a prior-itization criterion based on System Z. While it handles some benchmarks better than the individual systems do, its hetero-geneous character makes it hard to assess. It doesn’t share our goal to seek a plausibilistic semantics for abstract argu-mentation and seems to produce different results even in the generic context.
8
Conclusions
We have shown how the ranking construction paradigm for default reasoning can be exploited to interpret abstract ar-gumentation frameworks and to specify corresponding ex-tension semantics by using generic argument instantiations and distinguished canonical ranking models. We have con-sidered structured and conditional logical instantiations, de-fined attack between inference pairs, and after a further abstraction step, introduced simple semantic instantiations, which interpret arguments by triples of premise, strict, and defeasible content. While our basic ranking extension se-manticsEjz is intuitively appealing and has some
interest-ing properties, it also exhibits a surprisinterest-ingly unorthodox be-haviour. This needs further exploration to see whether there are approaches which share the same semantic spirit but can avoid abnormalities conflicting with the standard argumen-tation philosophy. Actually, we have been able to develop an alternative semantics which seems to meet these demands, but it will have to be discussed elsewhere.
9
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